From Hawthorne and Martin’s WBC commentary:

Who was this Clement? It is impossible to answer this question, other than to say that in all likelihood he was a Philippian Christian. Evidently he was so well known within the church that Paul did not need to identify him, and he did not think to do so for strangers who might chance to read his letter. Clement was a common Roman name. Hence to say that he was the Clement who later became the third bishop of Rome (cf. Eusebius, Hist. eccl. 3.4; 5.6) is simply to be making a guess. Such identification rests solely on agreement in name, not at all on historical evidence.